Abstract

Revisiting canonical integration of the classical solid near a hyperbolic or elliptic uniform rotation, normal canonical coordinates p, q are constructed so that the Hamiltonian becomes a function (“normal form”) of x+ = pq or of x− = p2 + q2: the two cases are treated simultaneously distinguishing them, respectively, by a label a = ±, in terms of various power series with coefficients which are shown to be polynomials in a variable depending on the inertia moments. The normal forms are derived via the analysis of a relative cohomology problem and shown to be obtainable without reference to the construction of the normal coordinates via elliptic integrals (unlike the derivation of the normal coordinates p, q). Results and conjectures also emerge about the properties of the above polynomials and the location of their roots. In particular a class of polynomials with all roots on the unit circle arises.